Physical & Interfacial Electrochemistry 2013Electrochemistry 2013

Lecture 2.Lecture 2.Electrolyte Solutions :

I / l t i t ti i i Ion/solvent interactions: ionic solvation.

Module JS CH3304 MolecularThermodynamics and Kinetics

Ionics: the physical chemistryWe recall that electrolyteof ionic solutions.

In previous lecture courses at the

We recall that electrolytesolutions are solutions which canconduct electricity. We recallfrom basic general chemistry thatp

Freshman level we presented arather general introduction toelectrochemical systems and

from basic general chemistry thatin solutions solutes can exist in anumber of possible forms:Molecular units: solution is non-concepts. In this lecture course we

begin the study of physicalelectrochemistry in earnest and setth b ll lli b di i I i

Molecular units: solution is nonconducting, normal colligativeproperties exhibited, XRDanalysis indicates discretethe ball rolling by discussing Ionics,

the physical chemistry ofelectrolyte solutions.

analys s nd cates d scretemolecular units in the solid.Molecular units plus ions: solution is weakly conducting, colligativey g gproperties indicate slightly more than expected numbers of particles present, XRD analysis indicates discrete molecular units in the solid.

We now begin the study ofphysical electrochemistry in

The term electrolyte is of particular importancehere. Now electrolytes may be defined as a class ofphysical electrochemistry in

earnest and consider thesolvation of ions in electrolyticsolutions . In particular in this

y ycompounds , which, upon dissolution in a polar solvent, dissociate at least partially into ions . If we zero inon the concept of dissociation , or the splittingapart to form ions , and concentrate on the extentp

lecture we shall focus on ion -solvent interactions , anddiscuss the progress that hasbeen made in the understanding

p ,to which dissociation takes place, then we candistinguish between two types of electrolytes .These two designations are termed true andpotential . A true electrolyte implies completebeen made in the understanding

of the nature and structure ofsolvated ions . Our mainemphasis will be on aqueous

p y p pdissociation into ions, whereas for a potentialelectrolyte only limited dissociation takes place .Thus sodium chloride NaCl is a true electrolytewhereas acetic acid CH3COOH is a potentialp q

solutions .3 p

electrolyte . These statements are well known frombasic freshman chemistry. The mechanism of ionformation is different for both of these electrolytetypes.In recent years many advances have

b d b th i i t l ypA slightly older designation which is still in current

use are the terms strong and weak electrolytes .Note that solutes giving in solution molecular unitsare termed non-electrolytes, those generating both

been made both in experimental techniques (spectroscopic and diffraction methods) and in computational simulation protocols y g g

molecular units plus ions, weak electrolytes andthose forming ions only, strong electrolytes.

computational simulation protocols (such as molecular dynamics MD, especially as applied to biologically important molecules) which may be

d t l f th t d f i i used as tools for the study of ionic solvation processes .

NaCl is an ionic compound consisting of Na+NaCl is an ionic compound consisting of Naand Cl- ions joined together via electrovalent bonds in a crystal lattice . When solid NaCl is put in contact with a solvent such as water , th l t l l idl tt k th the solvent molecules rapidly attack the lattice , disrupt and break the ionic bonds and thereby generate hydrated ions Na+(aq) and Cl-(aq) as outlined schematically below. Here (aq) as out n sch mat ca y ow. H r we see ion/solvent interactions very much at work.

Dissolution of crystalby action of solventby action of solvent.

Significant (complete)Dissociation/ionization.Ion/solvent interactionStrong.

A different situation pertains foracetic acid The latter is a neutral 2 3 ( )HA H O H O aq A acetic acid. The latter is a neutralorganic compound. In this case aspecific reaction between the soluteand solvent, a proton transfer

i l i h iWeak acidK small3H O A reaction, results in the generation

of acetate CH3CO2- and hydronium

ions H3O+ ions. A proton istransferred from the organic acid (a

KA small.3

[ ]AKHA

transferred from the organic acid (aproton donor) to a water molecule (aproton acceptor) outlined below. Wehave a Bronsted-Lowry acid/base

i N h lRefer to Bronsted-Lowry Acid/BaseB h i di d i JF CH1101reaction . Now the latter process

only proceeds to a limited extent .Typically the degree of dissociationis ca. 10-3 . Chemical method: proton transfer reaction.

Behaviour discussed in JF CH1101.

is ca. 10 .

Degree of dissociationsmall, 0.1 %.

The structure of water , the ubiquitous medium .

We see that when compared with other solvents, liquid water exhibits several distinctive properties For instance water q

Water is of course the most generally used solvent system in electrochemistry However for

distinctive properties . For instance , water has a high relative permittivity (er = 78) , high surface tension (72 mN m-1), high thermal capacity (75 J mol-1K-1) and thermal electrochemistry . However for

some applications non aqueous solvents must be used (e.g. in high density lithium battery systems).

conductivity (0.6 J s-1m-1K-1). Also , its molar volume is the smallest of all the common solvents and so it will exhibit the highest particle density A further point to note is As a general rule , an

electrochemically useful solvent must be able to dissolve a reasonable amount of an inorganic

particle density . A further point to note is that the enthalpy of vaporization is greater that the enthalpy of fusion in contrast to the situation observed for non polar solvents reasonable amount of an inorganic

electrolyte (ca. at least 10-3 mol dm-3) .

where the latter quantities are of similar magnitudes . The surface tension of water is also considerably larger than other solvents. The same can be said for the molar heat The same can be said for the molar heat capacity . Finally , considering the magnitude of the molar mass of water , we note that its viscosity is anomalously high. Hence these Bockris & Reddy, Modern

Electrochemistrybulk properties of liquid water suggest that the latter must be quite highly ordered and that considerable molecular association is present Furthermore we can conclude that

Electrochemistry2nd edition, Kluwer, New York, 1998, Vol.1 Ionics, Ch.2.

present . Furthermore we can conclude that intermolecular attractive interactions are stronger in liquid water than in other solvents .

The structure of water and icedepend on the geometry exhibitedb i l t l lby a single water molecule .Molecular structure determinesmacroscopic properties . We knowfrom Freshman Chemistry that afrom Freshman Chemistry that asingle H2O molecule has eightvalence electrons (six from oxygenand one each from the two hydrogent ) Th d d f i di id latoms). The demands of individual

atomic valencies dictate that theoxygen atom is covalently bonded toeach of the hydrogen atoms andeach of the hydrogen atoms andthat two lone pairs of electronsreside on the oxygen atom . Hence inthe language of quantum mechanics ,th t i 3 h b idi dthe oxygen atom is sp3 hybridised .Water exhibits a tetrahedralstereochemistry . The HOH bondangle is 104.5 o . The molecule actsg mas a dipole and the lp electrons mayenter into hydrogen bonding withhydrogens on neighboring waterm l l s In this thmolecules . In this way a threedimensional network structure maybe generated .

Methods of structural analysis such as

And what of the structure of ice ? The best experimental evidence suggests that the latter material Methods of structural analysis such as

XRD, ND, NMR, IR and Ramanspectroscopy applied to liquid waterhave indicated that under normal

diti 70% f t l l

suggests that the latter material consists of a network of open puckered hexagonal rings joining oxygen atoms together . Each oxygen

conditions ca. 70% of water moleculesexist in "ice floes" (clusters of ca. 50molecules exhibiting a structure similarto that of ice) . The mean lifetime of

atom is tetrahedrally surrounded by four other oxygen atoms . Also, in between any two oxygen atoms is located a hydrogen atom which is to that of c ) . h m an f t m of

such networked structures is ca. 10-11 s.

located a hydrogen atom which is associated to the adjacent oxygen atoms via hydrogen bonding . Hence each oxygen atom has two hydrogen atoms near it at an estimated distance of ca. 0.175 nm . Now such a network structure of associated water molecules contains interstitial regions molecules contains interstitial regions located between the tetrahedra , which typically are larger than the dimension of a water molecule . Hence it appears that a free non-associated water molecule can enter the interstitial regions without generating any disruption of the network any disruption of the network structure .

We can therefore say that liquid waterunder most conditions is described in terms

It is important to note that this distinction between network and free water is not a static one . Networks can break down :

of a somewhat broken down, slightlyexpanded form of the ice lattice . In short,liquid water partly retains the tetrahedralbonding and resultant network structure

more free water molecules can be formed . Free water molecules may combine to form clusters ( typically ca. 50 water molecules joined together) . The situation is g

characteristic of the crystalline structureof ice . Hence by analogy with the icestructure one has associated network waterand structurally free non associated water

j g )therefore quite dynamic . Free water can be transformed into network water and vice versa . This picture (icelike labile clusters and monomeric water molecules) was y

located in the interstitial regions in thenetwork .Note that the mean oxygen-oxygen bond distance in ice is 0.276 nm whereas in liquid

)initially proposed some time ago by Frank and Wen .

qwater it is 0.292 nm . Furthermore, the number of oxygen nearest neighbours in ice is 4 and in water is between 4.4 and 4.6 .

One expects that solvents such asMeOH and formic acid should also

How many water molecules must be joined togetherbefore the bulk properties of water are observed ?Put another way , we ask how many water moleculesMeOH and formic acid should also

exhibit considerable association viahydrogen bonding and that a certaindegree of order should be present .

y , ydoes it take before the cluster shows theproperties of wetness or before the dielectricproperties approach that of bulk water ? This is nota philosophical question : the answer leads one tog p

However as noted by Desnoyers andJolicoeur 2 the details of solvent-solvent interactions in polar liquids isgenerally unknown at the present

p p qthe forefront of research . A single water moleculewill not suffice : this much we know . A recenttextoook author has quoted a value of ca. 6molecules . This is clearly far too low .generally unknown at the present

time , water being the only exception. Solvent-solvent interactions canand do play an important part in ionic

ySimonson has recently calculated using

Molecular Dynamics (MD) techniques a value for thestatic dielectric constant of water using a simplemodel system consisting of a microscopic droplet ofp y p p

solvation . However their effect isvery difficult to establish in non-aqueous solutions . In this course wewill concentrate mainly on water

y g p pwater in a vacuum . The idea behind this approach isas follows .

Almost all computational studies to date have been based on simulations of bulk water either will concentrate mainly on water ,

since the latter has been wellcharacterised experimentally and isbest understood .

using so called periodic boundary conditions (more about these later on) or by combining a finite spherical microscopic region with an outer bulk region modelled as a dielectric continuum. Such

Solvent / solvent interactions caninvolve dipole / dipole , dipole /induced dipole , induced dipole /induced dipole (dispersion) and

gcomputations are difficult, time consuming and hence very expensive to do . Lots of long range interactions have to be taken into account and this causes convergence problems when the computation induced dipole (dispersion) and

dipole/ quadrupole interactions .

p pis done . Optimal computational strategies are still being pursued .

On the other hand using a modelsuch as a spherical water droplet

We now move on and consider somesimple approaches used for quantifyinsuch as a spherical water droplet

imbedded in a vacuum , one candispense with long rangeinteractions and one can readily

simple approaches used for quantifyingwhat happens when an ion enters fromthe gas phase into a polar solvent such aswater and becomes solvated (ory

compute all the interactionswithin the droplet whose size canbe chosen . Simonson has shownthat very good agreement (e in

(hydrated). In particular we shall considerthe energetics of the interactionbetween an ion and the solvent .

We shall initially adopt a verythat very good agreement (er inthe range 77 - 87) with theexperimental bulk dielectricconstant of water is obtained for

We shall initially adopt a verysimple first order approach and assumethat the solvent medium does not have amolecular structure , but instead can be

a droplet of 1963 TIP3P watermolecules (sphere 2.4 nm radius)at T = 292 K after a simulationtime of 1000 ps Hence from this

,regarded as a dielectric continuum . Inthis approach we assume that theinteractions between the ion and thesolvent are electrostatic in origin and wetime of 1000 ps . Hence from this

work it is clear that a cluster ofsome 2000 water molecules canmimic the behavior of the bulk

solvent are electrostatic in origin and weview the ion as a charged sphere ofradius r and the solvent as a dielectriccontinuum of dielectric constant er

liquid ..

r

The solvent is the medium in which the solute exists. It is often calledA dielectric. A dielectric can be thought of in terms of an insulator, which isA substance which stops or tends to stop the flow of charge., in other words to stop a current passing through it.If a substance which acts as an insulator is placed between two charges itReduces

• the field strengthh f i b h h• the force acting between the charges

• the electrostatic potential energy of interaction between the chargesAnd the factor by which it reduces these quantities is called the

l ti itti it relative permittivity r.It is important to note that this definition of relative permittivity is independent of any assumption as to what the dielectric is made of. In particular it is independent of any assumption that the dielectric is In particular it is independent of any assumption that the dielectric is composed of atoms and molecules, and so requires no discussion of the medium at a microscopic level.In effect the dielectric constant is just a proportionality constant In effect the dielectric constant is just a proportionality constant characteristic of the medium.

Solvation: Solute-solvent Non structural treatmentinteractions.

Introduction

Non-structural treatment.This is based on viewing thesolvent as a structurelessdielectric continuum into whichIntroduction.

We focus attention on chargedsolutes (ions) in polar solvents

dielectric continuum into whichthe ion is embedded as a chargedsphere. The methodology ofclassical electrostatics is used insolutes (ions) in polar solvents.

Much work has been done in thisarea of Physical Electrochemistryboth from a theoretical and an

classical electrostatics is used inthe detailed analysis of theenergetics of the problem.The energetics of salvation isboth from a theoretical and an

experimental viewpoint. The topic isalso of considerable importance inbiophysical chemistry when the

he energet cs of sal at on sexpressed in terms ofthermodynamic cycles and thepertinent thermodynamicp y y

solvation of macromolecular proteinmolecules is addressed.Firstly with respect to the

p yquantities such as GS, HS andSS are determined in thecontext of simple electrostaticp

theoretical analysis we considertwo distinct approaches tosolute/solvent interactions.

models.

The non-structural approach is Structural treatment.ppbased on the original modeldeveloped by Born (1920). Thistype of analysis is very

This defines the modern approach.Here we view the solvent as amolecular entity with a well defined

approximate but is still being usedwith some success.Good agreement between theory

d i i b i d i h

structure (as indeed it has). We needa model of the solvent in theimmediate vicinity of the ion. Againh l i iand experiment is obtained via the

introduction of empirical fittingparameters which serve to

tif th i i di s i th

the solvation energetics arequantified in terms of thethermodynamic entities GS, HS andS Th th ti l l sis is iquantify the ionic radius in the

solvent.The non-structural Born approachhas been generalised in recent

SS. The theoretical analysis is againbased on electrostatic models butthese are more elaborate and includeion/dipole and ion/quadrupolehas been generalised in recent

years to examine the salvationenergetics of macromolecules, andis so a topic of considerable

ion/dipole and ion/quadrupoleinteractions. The structuraltreatments based on the early workof Bernal and Fowler (1933) Theis so a topic of considerable

current interest in biophysicalchemistry.

of Bernal and Fowler (1933). Theresults obtained are encouraging buta fully satisfactory theoretical modelhas yet to be developedhas yet to be developed.

In contrast the experimentalanalysis of solvation is quite welly qdeveloped.Structural information on theenvironment of a solvated ion isobtained from:obtained from:diffraction methods: X-ray,neutron and electron diffraction,EXAFS etc.spectroscopic studies : uv/vis, ir, nmr, raman spectroscopies etc.Diffraction methods provide directinformation about the environmentinformation about the environmentsurrounding the ion in solution,whereas indirect structuralinformation is provided frompspectroscopic methods.

One difficulty is thatdilute solutions need to be examinedin order to focus in on ion/solventin order to focus in on ion/solventinteractions. If more concentratedsolutions are used then ion/ioninteractions come into play and cloudthe issue.

In this section we discuss an approach developed by Born which can be used to

d l th ti f i / l t In the Born approach a blend ofclassical electrostatics andmodel the energetics of ion/solvent

interactions in a convenient manner . This approach may have its origins over sixty years ago and is certainly limited, but it

classical electrostatics andthermodynamic cycles is adopted . Theobject of the exercise is to evaluate ,from first principles, the Gibbsyears ago and is certainly limited, but it

can , and has yielded a large number of qualitative and semi-quantitative insights . Now this approach is limited and has h d it d t t H thi

p p ,energy change GS associated withthe process of ionic solvation , whereone considers the transfer of an ionfrom a gaseous vapour at low partialhad its detractors . However this

continuum approach represents a simple, easily interpretable, and computationally inexpensive physical model that does not

from a gaseous vapour at low partialpressure to the desired solvent (i.e.water) in the absence of any ion/ioninteractions . In simple terms thenexpens ve phys cal model that does not

involve many adjustable parameters . The major problem with this approach is that the concept of a macroscopic dielectric

i li d t l l

pwork of transfer of an ion fromvacuum to a solvent is equated withthe Gibbs energy of solvation . Hencethe formidable task of quantifying theresponse is applied to a molecular

system. Max Born was one of the founding fathers of modern quantum mechanics

the formidable task of quantifying theenergetics of ionic solvation in astructured solvent is reduced usingthis approximate approach tof f m q m m

having proposed the probability interpretation of the wavefunction for which he ultimately obtained a belated N b l P i

pp ppanswering the question : what is thework done in transferring a chargedsphere from vacuum into a dielectriccontinuum ?Nobel Prize . continuum ?

Energetics of ionic solvation : Non-structural Treatment.Born Model (1920)Born Model (1920)

We consider the following situation. Initial state: no ion/solvent interactions. This corresponds to ion in vacuum Vacuum

IonThis corresponds to ion in vacuum.Final State: ion/solvent interactions operative.This corresponds to ion in solution.Thermodynamics states that for reversibleProcess taking place at const T & P

Vacuum Free energy changeGS

Process taking place at const. T & P, Free Energy change G equals net work (other than PV work) done on system.We need to calculate the work done inTransferring an ion from vacuum into a solvent

SolventCharged Transferring an ion from vacuum into a solvent.

A solvent such as water has a complex structure.To simplify our analysis we dispense with structure

Charged sphere

nucleus

Electroncloud

i iq z e

irirNet charge

and regard the solvent as a structurless dielectric continuum. This medium has a dielectric constant given by S The ion also has structure (ion + associated hydration sheath). We represent the ion

h d h f h d d

nucleus

as a charged sphere of net charge qi = zie and radius ri.

l

Dielectric constant S

Gibbs energy of solvation computed viaStructured solvent Structurless

Dielectriccontinuum

a thermodynamic cycle.

Born Model : Thermodynamic cycle to calculate the work done in transferring a charged sphere from vacuum into the solvent.

Main assumption:Only charge on ion isResponsible for ion/solventi t ti

Charged sphere in vacuumVacuum

interaction.Hence ion/solventinteractions are solelyelectrostatic in nature.

Remove charge Work of discharging

h d h

WD

SG

Uncharged sphere in vacuum

Zero electrostatic work of transfer

W = 0

The following thermodynamic cycle isdeveloped. The ion i , represented as acharged sphere of radius R,is

id d i t b i iti ll l t d i

Uncharged sphere in solvent

SolventWT = 0considered in to be initially located in a

vacuum, and the work Wd required tostrip the ion of its charge q = zie isdetermined . Then this unchargedsphere is transferred into the solvent

Charging process Work of charging WC

pdielectric medium . We assume thatthis transfer process involves no work .Then the charge on the sphere insidethe solvent is restored to its full valueq = zie and the work done in charging

Charged sphere in solvent

q = zie and the work done in chargingWc is determined . Finally, the ion istransferred from the solvent back intovacuum . The work done in thistransfer process is - Gs .

In a thermodynamic cycle the algebraicSum of all the work terms is zero.

From fundamental electrostatics we recall that the electrostatic potential at a certain

dcs

sctrd

WWGGWWW

0 point in space is defined as the work done

to transport a unit positive charge from infinity to that specific point . Now the electrostatic potential at a point located a

h h ( ) We now evaluate the work done in charging a sphere of radius R from q = 0 to q = zie , where zi denotes the valence of ion i and e is the fundamental electronic charge . To do h h h h

distance r from a charged sphere (r) is given by the product of the electric field strength E(r) (which defines the electric force per unit charge) acting on the moving

h d h d h h this we start off with an uncharged sphere and add infinitesimal amounts dq to the sphere until the final charge level is attained .The product of electrostatic potential and the h d h k

test charge and the distance r through which the charge is carried .

qdrqdrrErRR 1 charge increment dq gives the work increment

dw.

dw dq

R

drr

drrEr0

20 44

Direction ofIncreasingpotential

The total work done in charging the sphere is obtained by summing over all the little increments dw.

p

Direction of Electric fieldDirection of

transport of charge

PositivelyCharged sphere Unit positive

ezez

c

ii

dqdwW00

f gTest charge

dqRqdqRrW

ezez

c

ii

41

000

Total chargingwork Wc

Rezi

241 2

00 = permittivity of vacuum = 8.854 x 10-12 C2N-1m-2

Work done in discharging sphereis negative of charging work, Wd = - Wc

Note that in the expression for Gs there is anegative sign heading the rhs . Now since thes negat ve of charg ng work, Wd Wc

RezW i

d 241 2

0

dielectric constant r > 1 always and so 1/r < 1 thenGs will always be negative . The solvation proceedsspontaneously : ions prefer to be solvated than to bein vacuum . Hence solvation is thermodynamicallyf l h l d h h

The recharging work done when the sphere is present in the solvent dielectric is modified by the

favourable. The Born equation also predicts that thesmaller the ion (smaller R) and the larger thedielectric constant , the greater will be the solvationfree energy .d electr c s mod f ed by the

macroscopic dielectric constant of the medium r .

ez1 2

The Entropy and enthalpy of solvation are readily evaluated. Note that HS is a quantity readily compared with experiment.

R

ezWr

ic 24

1

0

Born expression for Gibbs energy of ezNG iA 22 1

sss STGH

idielc

vacds

ezWWG 111 2

Born expression for Gibbs energy of Solvation. TR

ezNTGS r

r

iA

P

ss

2

0

124

TezNH riA 22 11

rcds R 24 0

2 Molar GibbsEnergy of

TRH

rrs 2

0

124

A clear prediction is that the 2 2

0 0

1 11 14 2 4 2

iA As

r r

z eN N qGR R

Energy ofsolvation

A clear prediction is that thesolvation free energy shouldvary inversely with ion size R

We note that the Born modelmakes predictions whichsuggest that the ion solvent

Born Model : Comparing theory and Experimentgg

interaction is much strongerthan it actually is . We readilysee this if we compareexperimentally determined Hsvalues for the alkali metal ions

300

P.W. Atkins, A.J. MacDermott, J. Chem. Ed.59 (1982) 359-360

values for the alkali metal ionsand the halide ions with thatevaluated via the Bornexpression . If one plots Hsversus the inverse of theP li t l di t

200

250Born TheoryBorn TheoryRegression lineExperiment

Pauling crystal radius we notethat the linear relationshipbetween Hs and 1/R predictedby the Born expression is notobserved . The experimental

-H S 150

Li+N +p

data points are all much lowerthan the theoretical Bornequation line . In general theuse of ionic radii leads to anoverestimate of the solvation

50

100Na+K+,F-

Rb+Cs+

Cl-Br-I-overestimate of the solvation

energies of anions and inparticular the calculatedsolvation energy of cations canbe almost 100 kcal mol-1

t th i t l 10.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0

I

greater than experimentalvalues . In any event it is quiteremarkable that such a crudeelectrostatic model works atall ! In fact what is measured is

(R/)-1

Born theory suggests that ion/solvent interactions are much stronger than experiment shows them to be.

the enthalpy of solvation of asalt not that of an individualion .

Need to use effective dielectric constant of solvent region near ion rather than bulk value (ca. 80 for water).

Continuum Solvation Models: Recent Developments.

Rashin & Honig Model : J. Phys. Chem., 89 (1985) 5588-5593.In original Born model R was chosen as the ionic radius derived from crystallographic measurements. This leads to quitefrom crystallograph c measurements. h s leads to qu teconsiderable deviations between experiment and theory. Born Equation estimates (using crystallographic ion radii) of the enthalpy of solvation are considerably higher than values derived from experimental measurements. For example: f m p m n m u m n . F mpLi+ :R = 0.60Å, HS,calc = - 277.7 kcal/mol, HS,expt = - 146.3 kcal/mol.Na+: R = 0.95 Å. HS,calc = - 175.5 kcal/mol, HS,expt = - 118.9 kcal/mol.

Rashin & Honig suggested that R should be replaced by the parameter RC termed the cavity radius. Transferring the uncharged ion intothe solvent produces a cavity. The size of this cavity should be used in calculations involving the Born equation. The cavity radius g q yis defined as the distance from the nucleus at which the electron density of the surrounding medium becomes significant.

The cavity radius is a rather arbitary concept, but it has beeny y p ,suggested that the covalent radii of cations and the ionic radiiof anions provide a useful first approximation to the cavity radius.

B. Honig, K. Sharp, A-S Yang, J. Phys. Chem., 97 (1993) 1101-1109

iA

s RezNG

11

24

22Rashin & Honig Model : J. Phys. Chem., 89 (1985) 5588-5593.

ricR 24 0

2 2

20

114 2

iA rs

IC

z eN THR T

04 2 IC r rR T

Predicted and experimental Hs values are in good agreement, with the error typically being in the region of 7 % .n the reg on of 7 % .

The agreement between theory and experimentis quite good, the calculated values areconsistently larger than the experimental valuesconsistently larger than the experimental valuesby some 10 - 15 kcal mol-1.

Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.Bontha & Pintauro developed a modifiedB th d i l ItBorn thermodynamic cycle . It waspresumed that the total free energychange associated with the transfer ofan ion from vacuum to a polarizablesolvent was equal to the sum of the work Discharge work

Charged sphere in vacuum

Vacuumq

terms associated with the followingprocesses :• discharge of the ion in vacuum (Wd) ,• transfer of the neutral species fromvacuum into the solvert (Wt )

Discharge ProcessDischarge work

Uncharged sphere

WD

vacuum into the solvert (Wtr) ,• re-charging the species in the solvent(Wc),• restoring the solvent to its originalstate (WR)

transferring the charged ion back into

SG

Uncharged sphere in vacuum

WTTransfer Process

Transfer work

• transferring the charged ion back intovacuum .

The WR term is an important addition tothe Born approach . During the Uncharged sphere

WTTransfer Process

recharging of an ion in a polarizablesolvent it is to be expected that thesolvent dipoles are orientated in theelectric field generated by the ion . Thiswill result in a decrease in the potential Recharge process

Work of charging

Uncharged sphere in solventSolvent

WCwill result in a decrease in the potentialenergy of the solvent . Hence, when theBorn cycle is completed an amount ofwork WR must be added to the solvent inorder to restore the rotational andtranslational motion of the solvent

Charged sphere in solvent

CWork obtainedfrom solvent dipolerealignmenttranslational motion of the solvent

molecules i.e. the aligned solvent dipolesmust undergo relaxation . Dipole relaxation

process

Work to restoreSolvent toOriginal state

WR

g m

The net free energy of solvation Gs is given byAll four work terms can be calculated using

s D T c RG W W W W

W = work done in discharging ion in

All four work terms can be calculated using Classical electrostatics.Detailed analysis can show that

WD= work done in discharging ion in vacuumWT = work done in transferring neutral species to solvent

2

08DqW

aW ma c

pWC = work done in charging neutral species in solventWR = work done in restoring solvent to original state (relaxation of aligned

0 0

T

D

CV

W ma c

W dV E dD

original state (relaxation of aligned solvent dipoles).It was also assumed that the solventwas a polarizable dielectric medium,

20 0

0

r

r

rE

r rV V

dV E dE dV E d

was a polarizable dielectric medium, which implies that the dielectric constant varies with electric field strength and that dielectric saturation ff t t b id d

2

0

r

r

r

S rV

W dV E d

effects must be considered.Hence the dielectric constant near an ion will be much less than the dielectric constant of the bulk solvent 0

chargeelectric field vector, , electrostatic potential

electric displacement vectorr

qE ED E

constant of the bulk solvent

0 p

dielectric constant is function of electric field strength.r

r r E

The total Gibbs energy of solvation isOutlined across.

2

08

E

s T rqG W dV r E dE 0

0 08sVa

In order to evaluate the double integral, the interrelationship between the radially dependent between the radially dependent electric field and solvent permittivity surrounding an ion must be determined over the

22

0

3 1

r

r

r r

n

entire ion charging process.This is done by simultaneously solving the modified Laplace equation of the electrostatic

2

2

1coth

5 2

rr r n r

r r

n

equation of the electrostatic potential in a polarizabledielectric medium and the Booth equation which describes

22 B

nk T

optical refractive index of solventn the variation in the solvent permittivity with electric field strength.These equations (outlined

p = dipole moment of solvent molecule

k Boltzmann constantr = radial co-ordinate measured from centre of ion

B

These equations (outlined across) are quite complicated and must be solvednumerically using a finite

0 r rrdr a

Boundary Conditions

difference technique on a computer.

0 rdr r a = surface charge density C m-2

Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.

F l h f l l d l h d fFirstly, the variation of local dielectric constant with distance forvarious extents of surface charging in the range 40 to 100 % isoutlined. We note that the distance from the ion surface over

hi h di l t i t ti i b d t l th t thwhich dielectric saturation is observed gets larger the greater theextent of surface charging. The computed variation in the dielectricconstant of water (at 298 K) as a function of radial distance fromthe ion surface for two univalent ions with different radii (Li+ andthe ion surface for two univalent ions with different radii (Li+ andCs+) and for cations of similar radius and different charge (Na+,Ca2+ and Y3+) are presented. We also show the computed variationexpected using the one layer dielectric model of Abraham and Lizziexpected using the one layer dielectric model of Abraham and Lizziin these figures . The Abraham-Lizzi model predicts that thedielectric constant remains at its bulk value until a certain criticaldistance After that the value drops to a very low level Hence thedistance. After that, the value drops to a very low level . Hence theAbraham-Lizzi model does not differentiate between ions ofdifferent charges and size . It does approximate well the computedvariation in dielectric constant with radial distance for univalentvar at on n d electr c constant w th rad al d stance for un valentions of large radius , but only poorly approximates the variationexpected for small univalent and for all divalent and trivalent ions .

Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.

In table above we show the computed and experimental ionic solvationGibbs energies calculated using the Bontha - Pintauro model for a rangeof cations and anions in water at 298 K Goldschmidt values of the ionicof cations and anions in water at 298 K . Goldschmidt values of the ionicradii were adopted in the calculation . Agreement between theory andexperiment is quite good .

The analysis can also be extended to non aqueous solvents . In table above we show some results for three solvents : show some results for three solvents : methanol, 1,1-dichloroethane and acetonitrile . Again excellent agreement is obtained between theory and n w n y nexperiment . The model predictions are typically within 10 % of the experimental solvation energy values .gy

Results from theory agree quiteWell with experiment. TypicallyError is in region of 2=6%. This is aError is in region of 2 6%. This is agood result given nature of assumptionsmade (solvent is dielectric continuum).

What are the advantages of the BonthaWhat are the advantages of the Bontha-Pintauro model ? The model contains noadjustable parameters , and specificallyconsiders the effect of dielectric saturation inconsiders the effect of dielectric saturation inthe vicinity of the solvated ion . Agreementbetween theory prediction and experimentalmeasurements is quite good typically within 10measurements is quite good, typically within 10%. The Bontha-Pintauro approach is the mostrecent elaboration on the original Born modeldeveloped over 70 years ago , and may be usedde eloped o er 7 years ago , and may be usedas a convenient computational protocol toevaluate Gibbs energies of solvation . Howeverthe model is still firmly based on continuumyelectrostatics and does not view the solvent asa structured entity. This is a majordisadvantage , of this and all other Born typecontinuum electrostatic approaches .

Some elaborations on the structural theory of solvation.

A hi i i d llThe theoretical models discussedin the previous sections haveneglected the important fact that

At this point we introduce a centrallyimportant idea . We saw that watermolecules can be represented as dipoles

Hence one might expect that thewater has a very well definedstructure , and that this structurewill have a very large part to playin the solvation of a particular ion

. Hence one might expect that theintruding ions will interact with thedipolar hosts . This interaction is notgentle and causes major changes in thet t f t Th h i llin the solvation of a particular ion .

Details of water structure havebeen presented in a previousslide.We recall that in liquid waterh k f d

structure of water . The sphericallysymmetric electric field of the ion cantear water dipoles out of the waterlattice and make them point or orient

there are networks of associated(via hydrogen bonding interactions)water molecules and also a certainfraction of free unassociated

lattice and make them point or orientwith the appropriate charged endtowards the central ion .

fraction of free unassociatedwater molecules . We now proceedon and ask the question : whathappens to the structure of water

h h lwhen ions enter the solvent ?

Consequently, we arrive at apicture of ion / dipole forces asproviding the principal basis ofionic solvation . Because of theoperation of these ion/dipoleoperation of these ion/dipoleforces, we can imagine that acertain number of watermolecules in the immediatevicinity of the ion may be trappedand orientated in the ionic field .We arrive at a picture of ionsenveloped by a solvent sheath ofenveloped by a solvent sheath oforientated, immobilised watermolecules. These immobilisedwater molecules move with the

h h hion as it moves through thesolvent medium . The fate ofthese immobilised solventmolecules is tied up with that ofmolecules is tied up with that oftheir partner ion . The ion and itssolvent sheath form a singlekinetic entity .

What of the situation far away fromIn the region between the solvationsheath (where the electric field of theWhat of the situation far away from

the ion ? At a large enough distancefrom the ion, the electric field isnegligible and the solvent structure

ill b ff t d b th i i fi ld

sheath (where the electric field of theion determines the water orientation)and the bulk water (where theorientation of the water molecules is

i fl d b th l t i fi ld f thwill be unaffected by the ionic field .Here the bulk water structure willpertain . The solvent molecules will notrealise that the ion is there at all .

uninfluenced by the electric field of theion) , the orientating influences of theion and the water network operate in amicroscopic tug of war . The formerr a s that th on s th r at a . microscopic tug of war . The formertries to align the water dipoles parallelto the spherically symmetric electricfield of the ion, whereas the water

t k t i t k th tnetwork tries to make the watermolecules in this "in between" regioncontinue adopting the tetrahedralbonding arrangement required forbonding arrangement required formembership of the network structure .Hence caught between the two types ofinfluences, the water in this in between

i k t d t iregion seeks to adopt a compromisestructure that is neither completelyorientated nor disorientated . In thisintermediate region the water structurentermed ate reg on the water structureis said to be partially broken down asoutlined across.

Let us therefore summarise thesituation . We can describe three

i i (i) th i

We have noted that both the central ion and its immobilised layer of water

regions near an ion : (i) the primaryor structure enhanced regionlocated next to the ion where thewater molecules are immobilised and

ymolecules form a single kinetic entity . The number of water molecules associated with the ion during its translational motion is termed the water molecules are immobilised and

orientated by the electric field ofthe ion, (ii) the secondary orstructure broken region where the

l b lk t t f t i

translational motion is termed the primary solvation (hydration) number nh. We must distinguish the solvation number from the coordination number .

normal bulk structure of water isbroken down to various degrees , and(iii) the bulk solvent region wherethe water structure is unaffected by

The latter defines the number of solvent molecules in contact with the ion and is determined by geometric considerations the water structure s unaffected by

the ion and the tetrahedrally bondednetwork characteristic of bulk wateris exhibited . These regions areill t t d i th i lid

considerations .

It is important to note that solvation numbers determined via different experimental techniques can give illustrated in the previous slide.

It should be realised that theseregions differ in their degree ofsharpness . The primary solvation

experimental techniques can give different answers due to the ambiguity inherent in the definition of the solvation number concept . In contrast, p p m y

sheath is reasonable well defined .p

the in between region of secondary solvation is not as well defined . The regions of primary and secondary solvation define the region over which solvation define the region over which ion-solvent interactions operate .

We now examine a simple structural model invoking ion-dipole interactions which may be We set the ball rolling by assuming invoking ion dipole interactions which may be used to provide a quantitative estimate of the energetics of ionic solvation .

We set the ball rolling by assuming that the primary solvation shell occupies a certain volume corresponding to n primary solvent

We now present an account of a very simple simple model of ionic solvation in which the structural details (although described at a very basic level) of the

p g p ymolecules plus one more to make room for the bare ion . As a consequence of this supposition a volume corresponding to n+1 solvent described at a very basic level) of the

solvent is considered . This account will again dwell on the energetics of solvation and is based on the work reported many

volume corresponding to n+1 solvent molecules must be made available in the solvent for the immersion of a primary solvated ion . We therefore p y

years ago by Bernal and Fowler , Eley and Evans , and Frank and Evans . Again we resort to a simple thought

remove n+1 solvent molecules and transfer them to vacuum . A cavityis left in the solvent as a result of this process We let W denote the g p g

experiment (which is a favouritepursuit of physical electrochemists)to help clarify the situation . Theessentials are presented in across and

this process . We let WCF denote the work done in cavity formation . We assume of course that the size of an un-solvated ion is the same as that of essentials are presented in across and

follow the essential idea of Bernal andFowler.

a solvent molecule . This approximation is reasonable for some ions .

Bernal Fowler Model

Now before the n+1 solvent molecules The introduction of the primarycan orient around the ion in vacuum , thewater cluster must firstly dissociateinto n+1 separate molecules . We set WDas representing the work done in this

The introduction of the primarysolvated ion into the cavity causesdisturbance of the solvent structurein the immediate neighbourhood ofth l t d i A lt f thias representing the work done in this

dissociation process .Ion-dipole bonds are then

made between n solvent molecules and

the solvated ion . As a result of thisa structure breaking work term WSBmust also be included .

Finally we may ask is thethe target ion . A primary solvationsheath is formed . We must determinethe work WID done in this process . Inshort the interaction energy between an

F na y w may as s thstory complete . Has everythingbeen accounted for ? The answermust be no . What about our extras l t l l l ft i ?short the interaction energy between an

ion and a dipole must be evaluated .Then, the ion along with its

primary solvation sheath is transferred f h h h

solvent molecule left in vacuum ?This species must finally betransferred from vacuum into thesolvent . The work done in this

from vacuum into the cavity within the solvent . The work done in this case is equal to the Born enthalpy of solvation WB We note that the radius of the

process is Wc , the work ofcondensation .

WB . We note that the radius of the solvated ion R + 2rs is used in the Born expression previously derived .

Now this thought experiment or g pthermodynamic cycle may appear to be quite complicated , but it really is the most simple one that we may devise . More complicated situations may be p yenumerated as done by Bockris and Saluja . Implicit in our simple model is that the coordination number and solvation number are the same . Also we proposed that only the ion- dipole interaction need be considered . Both of these restrictive assumptions may be removed and more elaborate models developed . We have also not really considered the region of secondary solvation (the second "in between" layer in any specific detail , we merely y y p yspecify some work WSB . Again this limitation may be dispensed with in a more detailed analysis .

We set the ball rolling by assuming that the primary solvation shell occupies a certain

l di t i l t

Here we have differentiated between theelectrostatic terms WID and WB and theother work terms which have all beenvolume corresponding to n primary solvent

molecules plus one more to make room for the bare ion . As a consequence of this supposition a volume corresponding to n+1

l t l l t b d il bl i

other work terms which have all beenlumped together into a composite workterm W . We now must evaluate all ofthese work terms . The electrostatic workterms WB and WID are readily evaluatedsolvent molecules must be made available in

the solvent for the immersion of a primary solvated ion . We therefore remove n+1 solvent molecules and transfer them to

A it i l ft i th l t

terms WB and WID are readily evaluated .The former quantity is obtained directlyfrom the Born theory.

vacuum . A cavity is left in the solvent as a result of this process . We let WCF denote the work done in cavity formation . We assume of course that the size of an un-

l t d i i th th t f l t

p

r

rrs

iAB T

TrR

ezNW

20

2 1122

14

solvated ion is the same as that of a solvent molecule . This approximation is reasonable for some ions .

We can calculate the enthalpy of

Here R denotes the unsolvated ionic radius andrs denotes the radius of a water molecule .Hence the quantity R + 2r represents theWe can calculate the enthalpy of

solvation by adding together all of thecomponent work terms enumeratedpreviously .

Hence the quantity R + 2rs represents theradius of a primary solvated ion . In derivingthis equation we again consider a Born typetransfer process but in this case we are movinga primary solvated ion from vacuum into a cavity

BIDCSBDCF

CSBBIDDCFs

WWWWWWWWWWWWH

a primary solvated ion from vacuum into a cavitywithin the solvent . We also use the bulkdielectric constant of water in this expressionsince we assume for the sake of simplicity thatthe water structure outside the cavity is

BID

BIDCSBDCF

WWW

the water structure outside the cavity isnormal and undisturbed .

Th h l i i

Ion-dipole work.

In the R denotes the unsolvatedionic radius and rs denotes theradius of a water molecule . Hencethe quantity R + 2r represents the

The other electrostatic term is concerned with the interaction between the ion and the n free solvent dipoles in vacuum to form the primary solvated ion the quantity R + 2rs represents the

radius of a primary solvated ion .In deriving this expression weagain consider a Born type

vacuum to form the primary solvated ion in vacuum . This is a standard problem in electrostatics and the mathematical details are not difficult to follow . Our

bl i t l l t th i t ti g yp

transfer process but in this casewe are moving a primary solvatedion from vacuum into a cavitywithin the solvent We also use

problem is to calculate the interaction energy between a dipole and an ion placed at a distance r from the dipole center We assume that the dipole is within the solvent . We also use

the bulk dielectric constant ofwater in this expression since weassume for the sake of simplicity

center We assume that the dipole is orientated at an angle to the line joining the centers of the ion and the dipole . We assume that the dipole is of l th 2L

p ythat the water structure outsidethe cavity is normal andundisturbed .

length 2L .

rr

2L

y Py

r2

r

x

+ L-

L

r1 x + L-

LL

+ ++

Now the ion-dipole interaction energyUID is equal to the product of the

Also we note that

cosand222 rxryx UID is equal to the product of theionic charge qi = zie and theelectrostatic potential y(r) due to thedipole . We recall from basicl i h h h

cosand rxryx

. Hence eqn.2 reduces to 2electrostatic theory that the

potential due to an assembly ofcharges is given by the sum of thepotentials due to each charge

cos21cos22

221 r

LrLrrLLrr

(3)

potentials due to each chargeconstituting the assembly . Since thedipole consists of the charges + q and- q which are located at distances r1

d f h i P d fi i h

Inverting this expression we obtain 2/12

cos2111

LL (4)

and r2 from the point P defining theposition of the ion (see figure onprevious slide) , hence the totalelectrostatic potential is given by :

1

cos

rrrr

We now make the point dipole approximationand assume that P is located very far awayelectrostatic potential is given by :

210

114 rr

qr

(1)

y f yfrom the dipole and so r >> 2L . We can thenexpand the rhs of eqn.4 using the Binomialtheorem to obtain :

From basic geometrical considerationswe note that :

yLxr 222

cos283cos2

211

2

2

2

2

2

1

rL

rL

rL

rL

rr

xLLyxyLxr

yLxr

2222221

1

(2) .....cos2

1615

3

2

2

rL

rL (5)

If we now neglect terms of order higher thanL2/r2 we obtain :

In the latter p = 2Lq denotes the dipolemoment . We can replace the distance rL /r we obtain :

1cos32

cos11 23

2

21

rL

rL

rr (6)

pby the sum R + rs to obtain thefollowing expression.

p21 rrrr

In a similar manner we may show that :

cos)(4 2

0 srRp

(9)

We note from eqn 9 that the potential

1

cos2111

2

2/12

2

LL

rL

rL

rr (7)

We note from eqn.9 that the potential due to a dipole falls off approximately as the square of the distance r, whereas the potential due to a single point charge

1cos32

cos1 232

rL

rL

r

If we substitute eqn.6 and eqn.7 into eqn.1 we

p g p gvaries only as r-1 . This observation is readily explained . The difference in the behavior of the potential at large distances arises from the fact that the q q q

obtain the following expression for theelectrostatic potential at the ion due to thepresence of the dipole :

distances arises from the fact that the charges in a dipole appear close together for an observer located at some distance away from the dipole and their

cos4

cos4

2, 20

20 r

pr

Lqr (8)

y pfields cancel more and more as the distance r increases . We also note that the electrostatic potential also depends on the orientation angle on the orientation angle .

It is clear for maximum interaction we set = We can also look at more complex electrostatic interactions For and so cos = 1 and eqn. 9 reduces to

iepz

(10)

electrostatic interactions. For example we can look at the ion/waterQuadrupole interaction. The latter may be calculated using the

204 srR (10)

We assume that n solvent molecules d th i i th i h d ti

y gmethods of electrostatics and the following expression may be derived.

N nz e N nz epsurround the ion in the primary hydration sheath . Hence per mole of ions the ion-dipole work term is given by : 2 3

0 04 2 4A i A i

IQi s i s

N nz e N nz epWr r r r

204 s

iAID rR

epnzNW

(11) In latter expression represents theDipole moment and p is the quadrupolemoment and n is the coordination

This result may also be obtained in amathematically more rigorous mannerwhich we will not describe here but

moment and n is the coordination number.For water. The positive sign in second term onh d t ti dwhich we will not describe here but

involves Legendre Polynomials which areonly suitable for real black beltelectrochemists at the Graduate level!!!

rhs corresponds to cations andthe negative sign to anions. Hence in a more Sophisticated nalaysis we should Sophisticated nalaysis we should replaceWID by WIQ.

Ion-induced Dipole work

We also may have to account for ion/induced dipole interaction work WIID .This term arises from the fact that whenThe water molecule is in contact with the ion the electric field of the ion tends toion, the electric field of the ion tends todistort the charge distribution (quantified in terms of deformation polarizability in the water molecule and so induces a dipolepmoment on the water molecule.

2

2 404 2

iAIID

i s

n z eNWr r